This study presents Anthropological Facial Approximation in Three Dimensions (AFA3D), a new computerized method for estimating face shape based on computed tomography (CT) scans of 500 French individuals. Facial soft tissue depths are estimated based on age, sex, corpulence, and craniometrics, and projected using reference planes to obtain the global facial appearance. Position and shape of the eyes, nose, mouth, and ears are inferred from cranial landmarks through geometric morphometrics. The 100 estimated cutaneous landmarks are then used to warp a generic face to the target facial approximation. A validation by re-sampling on a subsample demonstrated an average accuracy of c. 4 mm for the overall face. The resulting approximation is an objective probable facial shape, but is also synthetic (i.e., without texture), and therefore needs to be enhanced artistically prior to its use in forensic cases. AFA3D, integrated in the TIVMI software, is available freely for further testing.

In this paper, we propose a novel formulation extending convolutional neural networks (CNN) to arbitrary two‐dimensional manifolds using orthogonal basis functions called Zernike polynomials. In many areas, geometric features play a key role in understanding scientific trends and phenomena, where accurate numerical quantification of geometric features is critical. Recently, CNNs have demonstrated a substantial improvement in extracting and codifying geometric features. However, the progress is mostly centred around computer vision and its applications where an inherent grid‐like data representation is naturally present. In contrast, many geometry processing problems deal with curved surfaces and the application of CNNs is not trivial due to the lack of canonical grid‐like representation, the absence of globally consistent orientation and the incompatible local discretizations. In this paper, we show that the Zernike polynomials allow rigourous yet practical mathematical generalization of CNNs to arbitrary surfaces. We prove that the convolution of two functions can be represented as a simple dot product between Zernike coefficients and the rotation of a convolution kernel is essentially a set of 2 × 2 rotation matrices applied to the coefficients. The key contribution of this work is in such a computationally efficient but rigorous generalization of the major CNN building blocks.

Geometrical and topological inconsistencies, such as self-intersections and non-manifold elements, are common in triangular meshes, causing various problems across all stages of geometry processing. In this paper, we propose a method to resolve these inconsistencies using a graph-based approach. We first convert geometrical inconsistencies into topological inconsistencies and construct a topology graph. We then define local pairing operations on the topology graph, which is guaranteed not to introduce new inconsistencies. The final output of our method is an oriented manifold with all geometrical and topological inconsistencies fixed. Validated against a large data set, our method overcomes chronic problems in the relevant literature. First, our method preserves the original geometry and it does not introduce a negative volume or false new data, as we do not impose any heuristic assumption (e.g. watertight mesh). Moreover, our method does not introduce new geometric inconsistencies, guaranteeing inconsistency-free outcome.

In this paper, we propose a novel formulation to extend CNNs to two-dimensional (2D) manifolds using orthogonal basis functions, called Zernike polynomials. In many areas, geometric features play a key role in understanding scientific phenomena. Thus, an ability to codify geometric features into a mathematical quantity can be critical. Recently, convolutional neural networks (CNNs) have demonstrated the promising capability of extracting and codifying features from visual information. However, the progress has been concentrated in computer vision applications where there exists an inherent grid-like structure. In contrast, many geometry processing problems are defined on curved surfaces, and the generalization of CNNs is not quite trivial. The difficulties are rooted in the lack of key ingredients such as the canonical grid-like representation, the notion of consistent orientation, and a compatible local topology across the domain. In this paper, we prove that the convolution of two functions can be represented as a simple dot product between Zernike polynomial coefficients; and the rotation of a convolution kernel is essentially a set of 2 × 2 rotation matrices applied to the coefficients. As such, the key contribution of this work resides in a concise but rigorous mathematical generalization of the CNN building blocks.

The proposed method of Boolean operations, BORES, is efficient and appropriate for virtual surgical planning. Moreover, it is simple and easy to implement. In future work, we will extend the proposed method to handle non-colliding components.